Location-Corrector for Removing Sun-Induced Effects in the Global Positioning System

1. Field of the Invention

This invention involves the correction of position information obtained from the global positioning system (GPS), and more specifically to the correction of GPS position information for errors induced by the GPS satellites orbit about the earth and the sun.

2. Description of the Related Art

The complete global positioning system (GPS) constellation involves twenty-four satellites separated by large distances, each possessing a highly accurate clock and a microwave source with precise frequency. The problem of transforming the frequency of coherent radio signals to the geocentric inertial coordinate frame (as required during the transfer of time information) involves an implicit, dependence on earth-sun-clock orientation that has been omitted, except in very-long-baseline interferometry (VLBI). To obtain the required time standard between satellites in the GPS system, each satellite in the system has an atomic (Cs or Rb) clock possessing a frequency that is accurate to one part in .about.10.sup.13 -10.sup.14. See, Copps, An Aspect of the Role of the Clock in a GPS Receiver, Navigation, Vol. 31, No. 3, pp. 233-242, 1984; Knable et al., Clock Coasting and Altimeter Error Analysis for GPS, Navigation, Vol. 31, No. 4, pp.289-302, 1984, which are hereby incorporated in their entirety by reference for all purposes. The orbit of each satellite is approximately circular (possessing an eccentricity of .about.10.sup.-3), has a 12 hour period and is located in one of three orbital planes. Each orbital plane intersects the plane of the equator at an angle of 55O. The signal from each GPS satellite includes a frequency offset .LAMBDA. (.about.1 part on 10.sup.10). This offset automatically accounts for important earth-induced gravitational blue-shift and second order Doppler effects (SDE's), which insure that GPS clocks are synchronized, based on the same procedure used in defining universal time (UT), and that GPS transmissions effectively are altered in a manner that would make them appear to originate from a frame that is stationary in the frame of the moving geoid defined by the surface of the ocean at the surface of the earth. As a consequence, time is being maintained and defined in effectively the same manner both in GPS satellites and through UT (as defined at the surface of the ocean).

The GPS is used to determine position by correlating information from coherent microwave transmissions provided by the constellation of satellites. See; Milliken et al., Principle of Operation of NAVSTAR and System Characteristics, Navigation, Vol. 25, No, 2, pp. 95-106, 1978 and Spilker, GPS Signal Structure and Performance Characteristics, Navigation, Vol. 25, No, 2, pp. 121-146, 1978, which are hereby incorporated in their entirety by reference for all purposes. The accuracy of this procedure is determined by non-systematic and systematic errors. Systematic errors are associated with known, appreciable effects that result from the inaccuracies of the frequency and of the length of the path of each signal from the satellite to a potential user. Non-systematic errors are inaccuracies in the determination of path length and frequency that can not be eliminated because they are the result of the known limitations of the existing procedures for determining these quantities. Optimal performance requires that non-systematic errors be minimized and systematic errors be eliminated.

An anomalous absolute error of 1-10 feet in user position relative to each GPS satellite exists as a consequence of an error in the determination of path length and/or frequency, and this absolute error varies periodically in the transmission from each GPS satellite as a function of time in a regular fashion: maximal and minimal errors from this effect occur approximately once every 12 hours. This anomalous error has been previously thought to be the result of an unknown ionospheric effect that has not been systematically removed by existing procedures. M. Weiss reported in Weiss, Apparent Diurnal Effects in the Global Positioning System, IEEE Trans. Instr. and Meas., Vol. 38, NO. 5, pp. 991-997, Oct. 1989, which is hereby incorporated in its entirety by reference for all purposes, a large, systematic variation in the Allan variance (See; Blair, Time and Frequency: Theory and Fundamentals, Nat. Bur. of Stds., Vol. 140, pp. 151-205, 1974, which is hereby incorporated by reference in its entirety for all purposes) between GPS time and UT (corresponding to a systematic variation of the normalized variance of the difference between measured clock frequencies provided by GPS and UT). These variations 1) occur in the transmissions provided by all satellites, 2) have their maximal peak-to-peak variation diurnally, and have maximal magnitude (when averaged over one-half day) corresponding to a deviation between GPS time and UT of 1-10 nanoseconds.

It has been assumed that orbital motion of the earth around the sun does not alter the frequency of the signal from a GPS satellite between the points of transmission and reception. As a consequence, no attempt has been made to incorporate known sun-induced variations in the frequencies of radio transmissions that have been identified from VLBI, as stated above. Because this effect is neglected in all GPS signals, the frequencies from all GPS transmissions as they are received will be different from the value that they are supposed to have by an amount that may be accounted for by a systematic correction that applies only during the time interval .DELTA.t required for the signal to reach the earth. A comparable trend in the Allan variance involving a regular, periodic variation, with period of one day (and maximal peak-to-peak variation occurring diurnally) can be inferred from the product of the maximum values of the time interval .DELTA.t.sub.max (.about.0.1 s) required for the signal to reach the earth and the fractional deviation in frequency .DELTA.f/f.sub.o .vertline..sub.max that results from the sun-induced effects outlined herein. This means the effect is not accumulative as it would be if it affected the accuracy of time-keeping and can be derived from the product of .DELTA.t with the deviation in frequency from the correct value. The resulting correction is periodic (with a period of one day) and has a maximal fractional magnitude of .about.10.sup.-8. Because the value of .DELTA.t is roughly 0.1 seconds, the resulting correction, on the average, for the difference between time "reported" by the GPS at the earth surface and UT is 1-10 nanoseconds during a twelve hour period. These deviations between UT and GPS time should be systematic, meaning that they should be seen in the reported time that is sent by all satellites.

The anomalous 1-10 foot error in the GPS has been determined not to be the result of ionospheric effect but occurs because of variations in the frequency of each of the two GPS L-band transmissions (for developing the C/A and P-codes) that are the result of known special and general relativistic effects (previously identified in very-long baseline interferometry (VLBI)) that result because each satellite orbits around both the sun and the earth. See; Chubb, Sun-Induced Variations in Time in the Global Positioning System, Astrophy. and Space Sc., Vol. 213, pp. 63-74, 1994, which is hereby incorporated in its entirety by reference for all purposes; Wiess, supra; Milliken et al., supra; and Spilker, supra. It is known from VLBI that because each satellite orbits in this fashion, in order to systematically account for changes in frequency occurring between points of reception and transmission, it is necessary to include the gravitationally and orbitally induced red- and blue-shifts in the frequency of GPS transmissions that result from the relative orientation and motion of each GPS satellite about the sun.

The objective of this invention is to provide an apparatus and method that allows a receiving station utilizing the global positioning system (GPS) satellites to determine its position with an accuracy of 10 cm to one meter.

Another objective is to produce an apparatus for improving estimates of GPS system accuracy for applications in the control segment in the determination of Kalman estimators, based on inputs related to satellite clock bias, frequency offset, and drift rate that incorporate sun-induced effects.

These and other objectives are achieved by a system that determines the systematic errors associated with sun-induced effects that alter the frequency of the transmission of every satellite within the global positioning system (GPS) because of effects that result because the GPS satellites orbit around both the sun and the earth. A correction factor for the systematic errors is determined and applied to the satellite transmission frequency to provide a true position within 10 cm to one meter.

FIG. 1 is a schematic showing the system for determination of location correction by removing the Sun-induced effects in transmissions of the satellites in the global positioning system.

FIG. 2 is a schematic showing the system for determination of location correction by removing sun-induced effects utilizing a computer.

FIG. 3 is a schematic of the system for determination of location correction by removing sun-induced effects utilizing a plurality of electronic circuits.

The subject invention is used to determine the systematic errors associated with Sun-induced effects that alter the frequency of the transmission of every satellite within the global positioning system (GPS) or in comparable navigation systems. The basic information utilized in the following formulation is derived from techniques well known to those skilled in the art. See, Kee et al., Wide Area Differential GPS, Navigation, Vol 38, No. 2, pp.123-143, 1991; Martin, GPS User Equipment Error Models.Navigation, Vol. 25, No. 2, pp. 201-210; Smith et al., Sensitivity of GPS Acquisition to Initial Data Uncertainties, Navigation, Vol 31, No. 3, pp. 220-232; Robbins, Reference Trajectories from GPS Measurements, Navigation, Vol. 35, No. 1, pp. 89-103, 1988; which are hereby incorporated in their entirety by reference for all purposes; and Milliken, supra.

Table 1 shows the symbology applied in the formulation set forth below.

To determine these systematic, errors, the position of each satellite relative to the center of the earth is denoted by r.sub.sa.sup.(i) =r.sub.sa.sup.(i) r.sub.sa.sup.(i), (i=1, 24), where r.sub.sa.sup.(i) is a unit vector pointing along the direction of r.sub.sa.sup.(i) and r.sub.sa.sup.(i) is its magnitude. Also, relative to the center of the earth, the ratio of the velocity of the ith satellite to c is .beta..sub.sa.sup.(i), where c is the speed of light. The ith satellite has angular velocity .OMEGA..sub.sa.sup.(i) =.OMEGA..sub.sa.sup.(i) .THETA..sub.sa.sup.(i) about its earth-centered axis of rotation, which is pointed along the unit vector .theta..sub.sa.sup.(i) and has magnitude .OMEGA..sub.sa.sup.(i). Its angular velocity is .OMEGA..sub.es =.OMEGA..sub.es .theta..sub.es ; and .theta..sub.es is pointed along the direction of its orbital axis about the sun (23.degree. towards the equator at the north pole relative to the polar axis). The associated position of the center of the earth is r.sub.es =r.sub.es r.sub.es where r.sub.es .about.9.2.times.10.sup.7 miles, and r.sub.es is a unit vector pointing from the center of the sun to the center of the earth.

In a non-circular orbit, r.sub.es varies between .about.9,2.times.10.sup.7 and 9.3.times.10.sup.7 miles. For both the circular and non-circular cases, the position of the ith satellite is r.sub.sas.sup.(i) =r.sub.sa.sup.(i) +r.sub.es. The red-shift expressions for earth-based receivers and transmitters are identified by substituting relevant values for a fictitious satellite moving in the frame of an observer on the earth, defined by .OMEGA..sub.sa.sup.(i) =.OMEGA..sub.e .theta..sub.sa.sup.(i), .OMEGA..sub.e =2.pi./day, .theta..sub.sa.sup.(i) =z, and r.sub.sa.sup.(i) =r.sub.e cos(.theta..sup.(i)), where r.sub.e .about.6380 km is the radius of the earth, and .theta..sup.(i) is the latitude of the reference frame. Relative to the center of the sun, the total velocity, v, divided by c(=.beta..sub.sas.sup.(i)) of the (i) transmitter or receiver located at r.sub.sas.sup.(i) may be approximated using ##EQU1## Here, dr.sub.es /dt is the rate of change of the earth-sun separation. In the most general case, included herein, dr.sub.es /dt.noteq.0. Relative to the center of the sun, the total velocity of the receiver or transmitter divided by c(=.beta..sub.sas) is .beta..sub.sas =.beta..sub.es +.beta..sub.sa since the relativistic corrections to this expression enter with order v.sup.3 /c.sup.3, and only terms of order v.sup.2 /c.sup.2 will be retained. From the equivalency principle, it follows that the observed frequency at the receiver, f.sub.2, located at r.sub.sas.sup.(2), of the signal, f.sub.1, that is transmitted from the satellite position, r.sub.sas.sup.(1), is obtained from a linear order Taylor expansion of the expression ##EQU2## in terms of its dependence on the gravitational potentials .phi.(r.sub.sas.sup.(1)) and .phi.(r.sub.sas.sup.(2)). In equation (2), n.sub.2 and n.sub.1, respectively, are unit vectors at locations 1 and 2 that point along the path of the signal. In absence of multipath effects (a valid assumption when the receiver is designed in a manner consistent with what is necessary for successful operation; See, Spilker, supra, pg.128) these two vectors may be equated. Also, f.sub.0 is the frequency of the microwave signal at the point of transmission. Because .beta..sub.sas.sup.(i) .about.10.sup.-4, it is sufficient to retain terms of order .beta..sub.sas.sup.i) 2 in Equation (2). Then, it follows that the fractional change in frequency .DELTA.f/f.sub.0 =(f.sub.2 -f.sub.0)/f.sub.0 is given by

where D.sup.1.sub.1.fwdarw.2 and D.sup.2.sub.1.fwdarw.2 are first order and second order Doppler effects (SDE's) and P.sub.1.fwdarw.2 is the red-shift. ##EQU3## is a portion of the "conventional" SDE contribution D.sub.1.fwdarw.2.sup.2 (GPS)(=W+S.sub.1.fwdarw.2.sup.2) that is already included as a correction to the GPS signal.

Neglecting, (1) the terms of order r.sub.21.sup.(i) /r.sub.es.sup.3 .about.10.sup.-4 r.sub.es.sup.-2 (which enter the expression .DELTA.f/f.sub.0 with error 10.sup.-16), and (2) the mass of the earth (M.sub.e) and satellite mass relative to the mass of the sun, one finds that ##EQU4## Here, P.sub.1.fwdarw.2.sup.e =-GM.sub.e /c.sup.2 [(r.sub.sa.sup.(1)).sup.-1 -(r.sup.(2).sub.sa).sup.-1 ] where P is the red-shift due to the earth.

In earth-fixed coordinates, the direction of the axis of rotation of a receiver or transmitter in the ith GPS satellite is defined by:

(1) the longitude .lambda..sub.GPS.sup.(i) of its intersection with the equator at the surface of the earth and

(2) the angle .alpha..sub.I .apprxeq.55.degree. between its orbital plane and the plane of the equator through the relationship

This is then applied to the original estimated position of the receiver by obtaining a correction .delta.f.sub.i /f.sub.0 to the frequency of the transmission of the satellite, from which a range correction .delta.D.sub.i =.delta.f.sub.i D.sub.i /f.sub.0 to the range d.sub.i is obtained, where D.sub.i is the initial estimate of the distance of the observer to the ith satellite. Using this estimate of the correction to D.sub.i, a new estimate, D.sub.i.sup.new, of the range is obtained using D.sub.i.sup.new =D.sub.i +.delta.D.sub.i .epsilon., where .epsilon. is an adjustable parameter, such that 0<.epsilon.<1. Using this new trial estimate of D.sub.i.sup.new of range, a new receiver position is estimated. This new estimate of receiver position is then used in place of the initial uncorrected estimate. Also, using the computed values of .delta.D.sub.i, the receiver covariance matrix, defined by the equation cov.delta.X.sub..mu. =(G.sub..mu..sup.T G.sub..mu.) G.sup.T.sub..mu. cov.delta.D[(G.sup.T.sub..mu. G.sub..mu.).sup.-1 G.sub..mu..sup.T ] is constructed where the sun-induced covariance matrix cov6D (associated with the errors from the sun-induced effect) is the 4n.times.4n matrix defined by the block diagonal form

m=1, . . . , n, where n is the number of satellites used (n=4 is the standard case), .delta..sub.i.sup.j =kronecker=0 if 1.apprxeq.j and =1 if i=j, and G.sub..mu. and G.sub..mu..sup.T are defined as in Milliken, supra. The new estimate of position is then used to compute a new estimate of position which is then used to compute a new sun-induced correction. The new correction is then used to estimate a new user covariance matrix cov.delta.X.sub..mu.. The quantity E.sub.r =.SIGMA..sub.i (cov.delta.X.sub..mu.).sub.ii provides a measure of the error. When E.sub.r <1 meter, a satisfactory estimate of the correction has been obtained.

Referring to FIG. 1, in the preferred embodiment 10, a plurality of satellites 12, within the electromagnetic view of a receiving station 14, electromagnetically transmit satellite ephemeris information and ionospheric correction information. The receiving station 14, in addition to determining an initial terrestrial location for the receiving station 14 determined from an initial estimate of the frequency of the satellites based upon the uncorrected GPS signal. See, Van Dierendonck et al., The GPS Navigation Message, Navigation, Vol. 28, No. 2, pp. 147-165, 1978, which is hereby incorporated in its entirety by reference for all purposes. The received data is further processed by a plurality of electronic circuits or a computer 16, as shown in the U.S. patent application 08/237,568, Gardner, APPARATUS AND METHOD FOR IONOSPHERIC MAPPING, which is hereby incorporated in its entirety by reference for all purposes, to obtain the satellites 12 latitude, longitude, speed, altitude. The processed satellite ephemeris data is then further processed by a plurality of electronic circuits or a computer 18, utilizing the source code similar to that shown in Appendix 1, to determine the velocity of the individual satellites 12, v.sub.sa.sup.1, and the position vector projecting along the line from the center of the earth to the individual satellites 12, r.sub.sa.sup.1. Also, the velocity of the earth in relation to the sun 19, v.sub.es, its separation, R.sub.es, and position, R.sub.es r.sub.es, by a plurality of electronic circuits or a computer 22, are determined. The receiving stations 14 velocity with relationship to the satellite 12, v.sub.sa.sup.2, and position, r.sub.sa.sup.2, are also computed by a plurality of electronic circuits or a computer 24. Utilizing the outputs of the electronic circuits or computers 18, 22, and 24, a plurality of electronic circuits or a computer 26 determines the unit vectors, n.sub.1 and n.sub.2, the velocities v.sub.es, v.sub.sa.sup.1, and v.sub.sa.sup.2, the quantities v.sub.sa.sup.1 .multidot.n.sub.1, v.sub.sa.sup.2 .multidot.n.sub.2, v.sub.es .multidot.n.sub.1, v.sub.es .multidot.n.sup.2, (v.sub.sa.sup.1).sup.2, (v.sub.sa.sup.1).sup.2, and (v.sub.es).sup.2. The output of the electronic circuits or computer 26 are applied to a plurality of electronic circuits or a computer 28 to calculate the sun-induced second order Doppler shift N.sub.1.fwdarw.2.sup.2 +O.sub.1.fwdarw.2.sup.2, and the change in frequency from which a correction to r.sub.sa.sup.2 is determined due to the sun-induced error. This correction is then applied to the plurality of electronic circuits or computer 24 where the initial estimate of the frequency obtained at the receiving station 14 to refine the estimate of the terrestrial location of the receiving station 14, thereby, providing position information accurate to within 10 cm to one meter, and which is applied to a display device 32, such as a computer or video display.

The operations described above may be accomplished on a computer, such as a Zeos Model PCI486DX4-100, manufactured by Zeos Data Systems Corp. of Minneapolis, Minn. or Mackintosh manufactured by Apple Computer Inc. of Palo Alto, Calif. Any other computer 34 may be utilized, as shown in FIG. 2, providing it has a means for input and control 34a and at least one adder 34b, subtractor 34c, multiplier 34d, divider 34e, comparator 34f, trigometric function generator 34g, memory capability 34h, averager 34i, and a means for displaying data 34j. Further, a plurality of electronic circuits, as shown in FIG. 3, having a capability of adding 36, subtracting 38, multiplying 42, dividing 44, comparing 46, generating trigometric functions 48 and averaging data 52 may be utilized to accomplish the aforementioned operations. The design of the foregoing electronic circuits is well known to those skilled in the art.

The input information required in Kalman estimators of the control segment for satellite position, satellite clock bias, frequency offset and drift rate, and satellite solar pressure constants per satellite may be improved by determining an improved estimate of satellite frequency at each receiving station employed in the control segment based on the incorporation of the change in satellite frequency at each receiving station by using the known position of each receiving station to determine sun-induced correction in the received signal at each receiving station to improve the accuracy of the estimates of the deviation of GPS time relative to the atomic standard time kept at each receiving station, as shown above. From the improved frequency measurements, an alternative estimate of the clock bias, frequency offset and drift for each satellite is obtained at each receiving station. These serve as inputs to the state vector used to determine Kalman estimators of satellite position, satellite velocity, solar pressure constants, and satellite clock bias, frequency, offset, and drift rate. See, Russel et al., Control Segment and User Performance, Navigation, Vol. 25, No. 2, pp. 166-172, 1978, which is hereby incorporated in its entirety by reference for all purposes.

The advantage of this invention is the improvement of position location within several feet, meaning that the greatest utility of the invention is in determining absolute position in situations where precise location (at the level of 10 cm to 1 meter) is required, such as approaches by aircraft to small landing areas under periods of extremely limited visibility or for remotely-steered vehicles in confined regions.

It will be understood by those skilled in the art that still other variations and modifications are possible and can be affected without detracting from the scope if this invention as defined in the claims. ##SPC1##